This lesson will introduce the methods of finding probabilities of two or more events happening.
In this tutorial, you're going to explore the ideas of probabilities that involve the word "and," versus probabilities that involve the word "or.” Specifically you will focus on:
In probability, sometimes you want to find the likelihood of two events happening at the same time, or the probability that two events happen consecutively from each other. This is called joint probability. It can be expressed as either A and B, or A, and then a symbol, with B. That symbol is called intersect.
Joint Probability/"And" Probability
The probability of two events A and B both occurring.
A and B, or A intersect B are both accepted notations.
Here’s an example.
On the roulette wheel, we want the event black and even.
For instance, the number two is a black sector and also even. Four and six are also black and even. Not all of the blacks are even numbers, like 29, and not all the even numbers are black, like 12. List them all out. B intersect E, black and even, are sectors two, four, six, eight, 10, 20, 22, 24, 26 and 28.
All the other evens are red, and all the other blacks are odd.
In a Venn diagram, we can represent "ands" with this middle section.
The numbers in the middle part mean they're in that even bubble and in the black bubble. They're both even and black. The other sectors fall somewhere else. The remaining evens 12, 18, 30, 34, 14, et cetera, are all even but not black. 11, 13, 15, 17, 31, et cetera, are black but not even.
All of the ones outside of the bubbles, zero, double zero, one, five, are neither black nor even, are odd, and are all red or green.
Sometimes you use the word "or" to say either this, or that, but not both. This is known as an exclusive "or."
I will have chicken or fish for dinner. This says I'm going to have chicken for dinner, or I'll have fish for dinner. But I'm not going to eat chicken and fish for dinner.
Do you want to buy these shoes or those shoes?
The inclusive "or" would include a statement like this-- I need a seven or a spade to win this poker hand. You could get a seven to win. Or you could get a spade to win. Or you could get a card that's both a seven and a spade.
You need to wear black, or a button down shirt to school today. Maybe someone's saying it's picture day and you need to wear a black shirt or a button down shirt. You might wear a button down shirt or a black shirt that isn't a button down shirt. Or you might wear a black button down shirt.
Here is a scenario about a waitress at a cafe. The waitress says to Paul, do you want coffee or tea, Sir? Paul says, coffee please. Would you like cream or sugar? Both, please. Notice the two "or's" here. Is this first "or" inclusive or exclusive? Is the second "or" inclusive or exclusive?
What you should have decided, was that the first one was exclusive.
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He's not going to order both tea and coffee. So she's giving him a choice of coffee or tea, but not both. The second one, on the other hand, is inclusive.
You can have cream in your coffee. You can have sugar in your coffee. Or you can get both in your coffee.
In probability, you always mean inclusive when you say "or." When you say events A or B, you mean either A, or B, or both. When you say even "or" black, you mean the ones that are black, or the ones that are even, or the ones that are both black and even.
"Or" Probability
The probability that at least one of two events, A or B, occur.
This idea of "or" actually encompasses three regions in the Venn diagram-- this region of even only, this region of black only, and this region of both. Even only, black only, or both-- all of those are in the event even or black.
This notation, E or B, we can also use this. This looks like an upside down intersect symbol, and it is. E union B. Union means putting them together.
Find the area in this that talks about grades and rural. Students were asked, what's the most important thing about you in school, versus their school location.
Are the grades in the most important thing? Is being popular the most important thing? Or is being good at sports the most important thing? Where are the students that are in rural schools and said grades?
They are there. What about grades or rural? This is any student who is rural, or any student who said that grades were the most important, or both.
So grades is all of the students on the top row. Rural is all of the students in the first column. The 57 are both of those two, the ones that said the grades were most important, and live in a rural area.
“And" probability requires that both conditions be satisfied, so that the outcome belong to both of the two events A and B. The notation is the word "and," or the intersect symbol.
"Or" probability requires that at least one of the events be occurring, so either A only, or B only, or both. We use the word "or," or the union symbol. And we can visualize both "and" and "or" probabilities in both Venn diagrams, and two-way tables.
And we did an example of each of those. So we talked about "and" probability. And we used the intersect symbol. We also talked about "or" probability, and used the union symbol.
Good luck.
Source: This work adapted from Sophia Author Jonathan Osters.
The probability of two events A and B both occurring.
The probability that at least one of two events, A or B, occur.